Construction of a curved Kakeya set

We construct a compact set in ${\mathbb{R}}^2$ of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a range of $p$, $q$. Moreover, our construction can be generalised from parabolas to a family of $C^2$ curves satisfying suitable curvature conditions.